Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 11-s + 12-s − 4·13-s + 16-s − 6·17-s + 18-s + 4·19-s − 22-s − 6·23-s + 24-s − 4·26-s + 27-s + 6·29-s − 8·31-s + 32-s − 33-s − 6·34-s + 36-s + 10·37-s + 4·38-s − 4·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.213·22-s − 1.25·23-s + 0.204·24-s − 0.784·26-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.174·33-s − 1.02·34-s + 1/6·36-s + 1.64·37-s + 0.648·38-s − 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(80850\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{80850} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 80850,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.171397752$
$L(\frac12)$  $\approx$  $3.171397752$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.96618037655633, −13.53441187469498, −13.05787973354798, −12.63437650988807, −12.02092441674805, −11.64577719154171, −11.10098890649860, −10.47298501751242, −9.964275190633980, −9.509226314833388, −9.013693376624080, −8.209337799576031, −7.908792508629649, −7.335488803257190, −6.703988316166782, −6.382062842272825, −5.550209400786189, −4.991591911808362, −4.569402783854540, −3.970518224066381, −3.299250698535926, −2.739567509526595, −2.136861676351148, −1.658429936123236, −0.4555697781279239, 0.4555697781279239, 1.658429936123236, 2.136861676351148, 2.739567509526595, 3.299250698535926, 3.970518224066381, 4.569402783854540, 4.991591911808362, 5.550209400786189, 6.382062842272825, 6.703988316166782, 7.335488803257190, 7.908792508629649, 8.209337799576031, 9.013693376624080, 9.509226314833388, 9.964275190633980, 10.47298501751242, 11.10098890649860, 11.64577719154171, 12.02092441674805, 12.63437650988807, 13.05787973354798, 13.53441187469498, 13.96618037655633

Graph of the $Z$-function along the critical line